Coordinate transformation

Toreducethenumberofunknownsinequation10itisnecessarytoexpress

S _i

^and

U _bi

^{as functions of U,V and W which are dened in the probe xed coordinate}

system,(

x _p , y _p , z _p

^{). Figure3denesthecoordinatesystemandtheanglesneeded}

to relatetheprobewirestothecoordinatesystem.

φ _i

^{istheangle betweentheprojectionof wireiin the(}

y _p − z _p

^)planeandthe

y _p

^{axis. Figure4showstheprojectionofthewires inthe (}

y _p − z _p

^)planeandthe

corresponding

φ

^angles.

If the wires are placed in a perfect triangle, the values of the angles will be

90 ◦ , 330 ◦

and

210 ◦

respectively. Twovelocitycomponentsaredenedinthe(

y _p −z _p

⁾

plane,

U _bi

^{is the binormal cooling of wire i and}

U _{T P i}

^{is the projection of the}

tangentialcoolingvelocityofwirei,tpreferstotangentialprojection.

U _{T P}

^and

U _b

canbecalculatedfortheindividualwires. TheyarefunctionsofV,Wand

φ _i

^.

U _{T P i} = V cos φ _i + W sin φ _i

⁽¹³⁾

U _bi = V sin φ _i − W cos φ _i

⁽¹⁴⁾

Thevelocitycomponentinthenormal-tangentialplaneofwirei,

S _i

^,isafunction

ofU andtheprojectionofthetangentialcoolingvelocity.

S _i ² = U ² + U _{T P i} ²

⁽¹⁵⁾

Substitutingfor

U _{T P}

^{inequation15yieldsS asafunction ofU,VandW.}

Figure4: Velocitiesandanglesin the(

y _p − z _p

^)projection

S _i ² = U ² + ( V cos φ _i + W sin φ _i ) ²

(16)

Theow angle

α

^{in thenormal tangentialplane must also bedened. From}

gureXanexpressionfor

α

^{iseasilyfound.}

Figure 5: Denitionof

α

α = arctan U _{T P}

U

= arctan

V cos φ _i + W sin φ _i U

(17)

Equations13,15and17maybesubstitutedintoequation10toyieldthenal

equationforthreedimensional owoverwirei.

U _i ² cos ² ( α _ei ) = ( U ² + ( W cos( φ _i ) − V sin( φ _i )) ² )

(18)

+cos ²

α _ei + arctan

W cos φ _i − V sin( φ _i ) U

+( W sin( φ _i ) + V cos( φ _i )) ²

Forthe three wireprobea set of three equations with three unknownsis

ob-tained.

Forthe individual wires the eective angle

α _e

^{must be found. This obtainedby}

placingtheprobeinauniformowwithavelocityS, andmeasuringtheresponse

formultipleyawangles,

α

^{,whilethepitchangle}

β

^{isheldconstantatzero.Equation}

8describestherelationbetweentheowvelocityandthevelocitymeasuredbythe

wire,U(E).TheratiobetweenSandU(E)canthenbefoundfromequation8.

U ( E )

S = cos( α _e + α )

U ( E ) cos( α _e )

⁽¹⁹⁾

Theresultsfrom measurementsat dierentangles

α

^{canthenbecurvttedto}

equation19byadjusting

α _e

^{toobtainthebestttothe}datapoints. Atypicalset of calibrationanglesis

α = − 20 : 5 : 20

. Insection 2.1.2

α _e

^{waspresentedasthe}

geometric angle betweenthe wirenormalin thenormal-tangentialplaneand the

x _p

^{axis. This is notentirelytrue,}

α _e

^{will also beafunction of the} properities of the individual wire and most importantly of the ow angle. The eective angle

approachassumesthat theeectiveangleisconstant,thisishowevernottruefor

large ow angles. A litterature review by Lekakis [5] found several estimates of

the limits for x-wireprobesranging from

± 12 ◦ − ± 20 ◦

. Russ and Simon found

that the range of valid angles were larger for three-wire probes, in the range of

± 30 ◦

(reportedinthelitteraturereviewofAanesland[1]).

2.2 Probe volume and frequency response

The spatialresolution isan importantproperty ofameasurementtechnique. All

measurementtechniqueshavealowerlimitforspatial resolution, the variationis

large. Thespatialresolutionofapitotequalsthediameteroftheprobeatleast,for

alaserdopplerthesizeofthecrossectionofthelaserbeamsisthelimit,inparticle

imagevelocimetryitwilldependonthewindowsizeandoverlappingamongother

factors.

ForasinglesubminiaturehotwireLigraniandBradshaw[6]foundtheideal

ra-tiobetweenwirelengthanddiametertobeapproximately

L/D > 260

for

L < 1 mm

for measurentsin aturbulent boundarylayer. Forlonger wires 'eddy averaging'

wasreported. Thewires usedin thisprojectis notclosetothedimensionsofthe

wiresusedbyLigraniandBradshaw,andcanthereforenotbeexpectedtoresolve

thesmallestscalesintheowaccurately. Thephysicalsizeofthethreewireprobe

willhoweverbeagreaterlimitingfactorthanthedimesionsoftheindividualwires.

Avelocitygradientacrossthemeasurementvolumeoftheprobewillmeanthat

thewiresintheprobeexperiencedierentvelocities. Inaowwithalargevelocity

gradient, i.e. close to awall, this can resultin large dierences acrosstheprobe

volume and distort the result. The size of theprobevolume will therefore limit

howlargegradientswhichcanbemeasured.

Theresponseoftheindividualwires isalso importantto obtainagoodresult.

Ifthefrequencyresponseofthehotwireanemomtersaredierent,someturbulent

componentscanbeoverestimated. If forexamplethegoalof anexperimentis to

validate whether aowis isotropicornot,adierencein frequency response can

responses.Toreducetheeectofthis,thesignalsshouldallbelteredatthesame

cutofrequency.

2.3 Turbulent pipe ow

A conned ow such as apipe ow will develop until a steady state solution is

reached. Assumingthattheowenteringthepipeisuniform, theboundarylayer

willimmediatelystarttogrowatthewall. Thenalsteadystatesolutionisreached

whentheinviscidcoreisgone,theowisthensaidtobefullydeveloped. Theform

ofthevelocityprolewilldependonwhethertheowisturbulentorlaminar,the

wallroughnessand thepressuregradient.

Fully developedturbulentpipeowwill exhibitcertaincharacteristics. Inthis

sectionabriefreviewofsomeofthese characteristicsisgiven.

2.3.1 The pressuregradient

Auniformowenteringapipewillberetardedbytheshearstressfromthewalls.

The pressure gradient will be greatest in the beginning, and gradually decrease

untiltheowis fullydeveloped. Atsteadystatethedriving forceof thepressure

gradientwillbalancetheshearstressonthewall.

∂P

∂x = τ _w 4

D

⁽²⁰⁾

Thewallshear stress canberelated to the wall-friction velocity,

u _∗

^{, which is}

animportantparameterin pipeow.

τ _w = ρu _∗ ²

⁽²¹⁾

Bycombiningequation20and 21thewall-frictionvelocitycanbe foundfrom

thepressuregradient.

u _∗ ² = ∂P

∂X D

4 ρ

⁽²²⁾

2.3.2 Mean velocityprole

Aturbulentpipeowwill consistofthreeregions.

•

^{Aninnerlayerclosetothewallwhere viscousshearisdominating}

•

^{Anouterlayerwhere turbulentshearisdominating}

•

^{Anoverlaplayermergingthetwolayerstogether,wherebothtypesofshear}

isimportant.

commonapproachisto identifytheimportantparametersinthedierentregions

and apply dimensional analysis. In the inner region the velocity is assumed to

depend on the wall shear, uid properties and the distance from the wall. Free

stream conditionsareassumed notto be important. The wallshear will however

dependonfreestreampropertiessuchasthepressuregradient.

u ¯ = f ( τ _w , ρ, µ, y )

(23) Dimensionalanalysisyieldstwodimensionlessparameters.

u ¯

Thetwodimensionlessgroupsaredenoted

u ⁺

^and

y ⁺

respectively,giving

u ⁺ = f ( y ⁺ )

. In the inner viscous shear dominated region turbulent shear can be ne-glected. Analysisof themomentum equation will thenyield that

u ⁺ = ( y ⁺ )

, see e.g. White[12]. Intheouterregionofthepipeowthevelocitynolongerdepends

on viscous shear, but on the freestream pressure gradient and the radius of the

pipe,R.

U _cl − u ¯ = f ( τ _w , ρ, R, ∂P

∂x ) , y

⁽²⁵⁾

Dimensionalanalysisyieldsthreedimensionless groups.

U _cl − u ¯

Somewherebetweentheinnerandouterlayer,thetwolayersmustmerge,giving

thesamevelocity. Atagivenaxialpositioninthepipe,theshapeofgisassumed

tobeafunctionof

ξ = ^R

τ w

∂P

∂x

^{. Theoverlaplawforagiven}

ξ

^{canthenbefoundby}

manipulating equation24and 26.

u ¯

Thetwo regionscanonly bemerged ifthe fand g are logarithmic functions.

Theresultingrelationcan bewritten bothintermsofinnerandoutervariabels.

u ¯

Dierent valueshavebeen suggestedfor theconstants

κ

^{and B,but theyare}

consideredto benearlyuniversial.A willdepend on

ξ

^.

In a fully developed pipe ow the only mean velocity component is that in the

streamwisedirection,U.Insection2.3.2itwasshowedhowUvariesasafunction

ofy. Theshearstressesin aowarecloselylinkedtothemeanvelocitygradients.

ThegeneralizedBoussinesqeddyviscosityhypothesissuggestsarelation.

u _i u _j = ν _T ∂U _i

∂x _j − 2

3 ρkδ _ij

⁽³⁰⁾

Basedonequation30,onecanmakesomassumptionsonthemagnitudeofthe

shear stressesin a pipe ow. Theonly mean velocity gradient is

∂U

∂y

^{, one would}

thereforeexpect

uv

^{tobethedominantshearstressintheow.}

The variation of

uv

^{as a function of y can be found from} manipulation of the Reynolds averagedNavier-Stokesequations. Equations 31 and 32 show the

simplied RANSequationsforthepipeow.

∂p

Byintegratingequation32withrespecttoy,from0toyanexpressionforthe

pressureatagivenycoordinateisfound.

P

ρ + ¯ v ² = P 0

ρ

⁽³³⁾

Ifonetakesthederivativeof thepressurewith respectto x onewill nd that

∂P

∂x

^{isconstantwithrespecty,seeingthat}

v ²

^{isnotafunction ofx.}

∂P

∂x = ∂P ⁰

∂x

⁽³⁴⁾

The equation in the x-direction can be integrated in the same manner, with

respect to y from 0 to y. By using the fact that

dP

dx

^{is constant the following}

expressionisfound. theknownsituation at thecenter linean expressionforthe variationofthetotal

stresscanbefoundasafunctionofy.

−uv + µ

Equation37providesvaluableinformationabouthow

uv

^{vary asafunction of}

y. In the inviscid region viscous shear stress is neglible and the turbulent shear

stress is expected to vary linearly with respect to y. And at the center line all

shearstressesareexpectedtobezero.

2.3.4 Turbulentnormal stresses

Boussinesq estimatesthe normalstressesto beonethird ofthe turbulentkinetic

energyk. Inturbulentpipeowthatisnotthecase. Equation30assumesisotropic

and homogeneous turbulence, but in a shear ow the production of the normal

stresses will vary. In the case of a turbulent ow the turbulent kinetic energy

equationin theaxial directionwillbetheonlyonewithaproductionterm.

P roduction = −uv ∂U

∂y

⁽³⁸⁾

Theproductiondependsonthemeanowgradient,asmeanvelocityin they

andzdirectioniszeroforafullydevelopedpipeowtheproductionoftheturbulent

normalstressesis zero. Thisdoesnotmeanthattheothernormalstresseswillbe

zero. Energy is transfered from

¯ u ²

^to

v ¯ ²

^and

w ¯ ²

^{by nonlinear} pressure-velocity interactions[8].

2.4 Cylinder wake

ThecylinderwakeisacomplexandReynoldsnumberdependentow. Forverylow

Reynoldsnumbers,Re<49,alaminar,symmetricalandsteadyrecirculationregion

is presentbehindthe cylinder. As the Reynoldsnumberincreaseslaminar vortex

shedding will begin. When the Reynoldsnumber reaches about 194 streamwise

vorticesbegintoform[4]. Uptoabout

Re _D = 1000

theStrouhalnumberincreases [11]. TheStrouhal numberis dened astheratio between

f D

^{and U, where fis}

thevortexsheddingfrequencybehindthecylinder.

St = f D

U

⁽³⁹⁾

For

Re _D > 1000

the Strouhalnumber startto decrease untill it stabilizes for

10000 < Re _D < 100000

atavaluecloseto0.21[11]. Theregionfrom

Re _D = 1000

to

Re _D < 200000

isnamedthesubrcriticalrange[13]. Inthesubcriticalrangethe boundarylayeronthecylinderremainslaminar. IftheReynoldsnumberincreases

furthertheboundarylayerstartstodevelopfromlaminartoturbulent,movingthe

pointoftransitionupstreamandtheseparationpointdownstream. Thisresultsin

reduceddragandanarrowedwake.

Unlike apipe ow, the wake is continually evolving. The mean velocity eld

will continue to developuntil free stream conditions are reached. Momentum is

continuallytransportedtowardsthecenterofthewakewherethevelocitydecitis

form yieldsthefollowing.

∂V

∂y = − ∂U

∂x

⁽⁴⁰⁾

The continuity equation tells us how the gradient of V with respect to y is

expected to vary. Far from the centerline

dU

dx

^{will benegative, sincethe wakeis}

expanding. Inthisregion

dV

dy

^{willbepositive. Closertothecenterofthewakewe}

expect

dU

dx

^{tobepositive,}

dV

dy

^{mustthereforebenegative.}

By performing an order of magnitude analysis, the x-direction Reynolds

av-eragedNavier-Stokesequation canbe simplied considerably. Thetwodominant

terms arethe U gradientwith respectto x and crossectionalgradientof the

tur-bulentshear stress

uv

^.

U ∂U

∂x = − ∂

∂y ( uv )

(41)

Fardownstream from thecylinder

( x/D > 80)

theowcanbeassumedto be self-preserving[ref], which means that theshape ofthe proleis preservedalong

thex-axis. Theshapeoftheprolecanbefoundbystartingwithequation41and

makingsomeadditionalassumptions.

3.1 The hot-wire probe

The probe consist of three wires on six supporting prongs, asshownin gure3.

Thechosengeometryisdenedbytwoproperties:

•

^{Theprojectionofthewires inthe}

y _p − z _p

^{planeisatrianglewith60degree}

angles

•

^{Thewires areinclinedanangle}

α _e = 35 . 26 ◦

relativetothe

y _p − z _p

^plane

Thesepropertiesgiveageometrywherethewires areorientatedperpendicular

to one another. The geometry is the same as recommended by Aanesland [1].

It was chosen to reduce the probe volume and give a good cooling response in

all directions. The probesare manufacturedto t theabovedescription,but the

angleswillneverbeexactlycorrect. Theeectiveanglesmustbefoundtroughthe

procedure described in section 2.1.4. By taking a picture of the probe trough a

microscope,theorientationofthewiresin the

y _p − z _p

^{planecanbefound, such a}

photo can be seenin gure 6. Theangle

φ 1

^{is used to relate therotationof the}

probein the

y _p − z _p

^{planetocoordinatesystem.}

Figure6: Pictureofthe

y _p − z _p

^{planetakentroughamicroscope}

Thelengthofthesupportingprongsischosensuchthattheowoverthewires

are notinuencedbytherest of theprobes. It isalso important that theprongs

are nottoolong,asthiscancausevibrationswhich inturn canbe interpretedas

aturbulentvelocitycomponent.

Aplatinum(90

%

),rhodium(10

%

)alloyisusedinthewire. Thisalloygives agoodoxidationresistance,relativelyhightensilestrengthbuthasarelativlylow

The diameter also inuences the sensitivity of the probe. In this project a wire

with d =

5 µm

^{is used. This is arelativley thick wire, which reduces} sensitivity but increases mechanicalstrength. Thelength ofthe wirebetween theprobesis

approximatly4mm,ofthis approximatly1.75mmof thecoatingonthewirehas

beenetchedawayin thecentre. Thisgives

l/d ≈ 350

. Toreduce theinterference ontheowfromthesupportingprongsthedistancebetweenthesupportingprongs

should notbetosmall,theshapeoftheprongtipswillalsoaect theow[2].

Thecrossectionofthemeasurementvolumeis

≈ 5

mm,andthespatial resolu-tionoftheprobeisthereforassumedtobe5mm.

3.2 Measurement chains

Figure7describesthemeasurementchainintheexperiment.

Figure7: Measurementchain

Thehot wireanemometersare optimized for

1 µm

^notfor

5 µm

^{which is used}

in this experiment. For the initial measurement setup a high frequency

distur-bance appearedonthesignalathighvelocities

( > 12 m/s )

. This isaresultofthe inability of the control circuitto regulate thewire voltage. It could to acertain

degreebehelpedbychangingthebiassetting ontheanemometer. Thisincreased

the dampingin the control loop at the cost of a lowerfrequency response. For

thepipemeasurementsthiswassucient,in thecylinder wakehoweverthelarge

uctuationsrequired that higher velocities couldbe measured. The solutionwas

to extendthecable, increasing

R _cable

^{,and therebyincreasing thedampinginthe}

loop.

3.3 Signal sampling rate

The sampling rate must be set accordingto the timescale of the smallesteddies

ofinterest. Whentherangeoftimescalesexpectedisunknown,thesamplingrate

mustbesetaccordingtothesmallesttimescaleoncanexpect. Kolmogorovsmicro

notbeendoneinthisproject.

Thelimitingfactorforthesamplingrateisinthiscaseisthefrequencyresponse

oftheanemometers. Thefrequencyresponsewasfoundtovarybetweenthewires

from approximatly6.3 kHzto 8.0 kHz. A lowpass ltercut o frequencyof 6.5

kHz waschosen. Thesamplingrate shouldbeset accordingto thesamplingrate

theoremorNyquistcriteria,whichstatesthatthesamplingrateshouldbegreater

thantwicethemaximumfrequencyexpectedtoavoidaliases[10].

A suitablesampling time should bechosensuch that repeated measurements

givethesameresult,averagingoverrelevanttimescalesintheow.

Inthecylinderwakemeasurementsasampling rateof 13kHzwasusedalong

with a sampling time of 20 seconds. For the pipe measurements the sampling

frequency wasset to 7 kHzand the sampling time to 10 seconds. This wasnot

intendedtobethenalmeasurements,but simplypreliminarymeasurements,the

reducednumberofdatapointsgavesignicantlyreduceddatasizeandwastherefore

chosenatthetime.

3.4 Data reduction program

The sampled signal from the velocity calibration, the eective angle calibration

and traverses, were stored in text les and imported into a Fortran script. The

script corrects the data for temperature change, ts polynomials to the velocity

calibration data, calculates the eective angles, and uses the calibrationdata to

calculatetimeseries ofvelocityvectorsfromthevoltagetimeseries.

The solution of the equations (Eqs. 18) was be found by using a zero point

nder. Initially afortranfunction called DNSQE from theSLATEC librarywas

used, this function had previouslybeenused byAanesland [1] with success. The

algorithmworkedneforaveragedvoltages,butconvergensproblemsarisedwhen

the turbulenttimeseries wereanalyzed. As analternativeMatlabs fzero function

wasused. TheMatlabfunctionisconsiderablyslowerthantheFortranroutinebut

it doesthe job. Simple constraintswere placed on the solutionto insure that a

physicallycorrectsolutionwasfound. TheFortranscriptusedfordataanalysisis

describedfurherinappendix6.

3.5 Pipe ow rig

ThepiperigconsistsofahydraulicallysmoothPVCpipe,with adiameterof 186

mm and a length of 83 diameters. Ten pressuretaps are mounted on the pipe,

making it easy to measure the pressuregradient. The pipe is tted such that a

traversecan be mounted ontop, making it possible to traverse the ow through

thecenterofthepipe. Velocitiesinthepiperigcouldbevariedfrom5to12.5m/s.

Thecoordinate systemusedin thepipehasits reference

( y = 0)

onthe centre lineofthepipe,yispositiveabovethecenterline,andnegativebelowthecentreline.

The velocities in thepipe aredenoted

U _x

^,

U _r

^and

U _theta

^{and arethe axial, radial}

andcircumferentialvelocitiesrespectively.

An openloop wind tunnel is used forthecylinder wakemeasurements. The test

sectionis45cmx45cmand110cmlong. Acylinderwithadiameterof47.5mm

is tted in the center of the test section, leaving 50 cm of distance downstream

for the ow to develop. Measurementsare takenat

x/D = 10

. Velocities in the windtunnelcouldbevariedfrom4to30m/s.

Inthewindtunnelthecentreofthewakeisthereference

( y = 0)

inthe coordi-nate system,yispositiveabovethecenterline,andnegativebelowthecenterline.

U,VandWaretheaxial,vertical,andtransversevelocitycomponentsrespectively.

4.1 Calibration and testing

4.1.1 Velocity calibration

Inthepiperigthevelocitycalibrationwasperformedforvelocitiesbetween5and

12.5m/s. Athird order polynomialt to thecalibrationdata includingthezero

velocitypointgavearesidualoftheorderof

10 − 1

,whileasecond orderttothe

velocitypointgavearesidualoftheorderof

10 − 1

,whileasecond orderttothe

In document Three-dimensional wake measurements (sider 16-0)